That Carl Bergstrom thread on herd immunity is a good read. I saw a lot of people that misconstrued the idea and what it means when people were going after Fauci a couple months ago.
It isn't an on/off switch. It is a probabilistic concept that will vary depending on the environment, the variants, and behavior. In the purest sense it means R<1 without any mitigation, such that an outbreak won't sustain itself and eventually a virus will recede. And in a basic model you get it as 1-1/R0. The issues are we don't exactly know what the R0 is, the new variants are much more transmissible than the original strain, and reality is a tad more complicated than the basic model such that NYC would need a higher herd immunity threshold than Lubbock, Texas, a dorm would need a higher threshold than an outdoor park, and say a retail employee would have a higher transmission rate than a retiree.
And as Bergstrom mentions, case levels matter. Reaching the HIT with cases high means overshooting the HIT as those cases transmit. That is why vaccine-derived herd immunity is preferable to infection-derived. And as he mentions, as we remove mitigation controls, the Rt will increase.
So ideally we want to be at a very high level of vaccination and get cases down to a low level. For Measles, the most contagious virus we know of with a R0 of perhaps up to 18, the herd immunity threshold is believed to be as high as 95% as we see sustained outbreaks even when vaccination rates drop down to the high 80s/low 90s. Coronavirus may not have a HIT that high, but the higher the vaccination rate the better as herd immunity is a local phenomenon.